Name: ________________________ I can solve one step equations using addition and subtraction I can solve one step equations using multiplication and division I can solve one step equations with fractional coefficients I can solve two step equations I can solve multi-step equations by combining like terms I can solve multi-step equations by using the distributive property I can solve multi-step equations with variables on both sides I can solve and graphing inequalities in one-step and two-steps Unit 2 Beaumont Middle School 8th Grade, 2017-2018 Introduction to Algebra
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Unit 2 Beaumont Middle School Name: 8th Grade, …fergusonmathbms.weebly.com/uploads/5/4/1/1/54113523/unit_2_pack… · We get an answer of 8. ... ~~ Unit 2, Page14 ~~ Homework is
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Name: ________________________
I can solve one step equations using addition and subtraction
I can solve one step equations using multiplication and division
I can solve one step equations with fractional coefficients
I can solve two step equations
I can solve multi-step equations by combining like terms
I can solve multi-step equations by using the distributive property
I can solve multi-step equations with variables on both sides
I can solve and graphing inequalities in one-step and two-steps
Unit 2 Beaumont Middle School 8th Grade, 2017-2018 Introduction to Algebra
~~ Unit 2, Page2 ~~
SOLVING EQUATIONS BY ADDING OR SUBTRACTING
When you solve an equation, the goal is to get the variable alone. The value on the other side
of the variable tells you the solution of the original equation. You use inverse operations,
which undo each other, to get the variable alone. (Remember that, in previous math classes,
you used related equations like 3 + 5 = 8 and 8 – 3 = 5. These equations show that addition and
subtraction undo each other.)
SUBTRACTION PROPERTY OF EQUALITY:
You can subtract the same number from each side of an equation.
Arithmetic Algebra
10 = 2(5) If a = b,
10 – 5 = 2(5) -5 then a – c = b – c
ADDITION PROPERTY OF EQUALITY:
You can add the same number to each side of an equation.
Arithmetic Algebra
10 = 2(5) If a = b
10 + 3 = 2(5) + 3 then a + c = b + c
EXAMPLES:
1) y + 5 = 13 2) c – (−12) = 24 3) x – 10 = 12
− 5 −5 c + 12 = 24 (ADD THE OPPOSITE!) x + -10 = 12
y = 8 − 12 −12 +10 +10
c = 12 x = 22
Check:
1) 2) 3)
PRACTICE.
a) a + 8 = 3 b) 5 = d + 1 c) c + (-4) = -5
d) y – 5 = 8 e) p – 40 = 42 d) 98 = x – 14
Objectives: I can solve one-step equations using addition and subtraction.
~~ Unit 2, Page3 ~~
Applying Solving Equations When translating words in sentences, you can write equations to be solved. This skill will enable you to solve more complex word problems. Let’s review some words and their corresponding math symbols. Make a list of words that correspond with the given symbol.
+ − * ÷ =
PRACTICE Write an equation for each sentence. You may have to choose your own variable. 1. The sum of 63 and some number, x, is -82. Find the number. ______________________________
2. Sixty-eight is ninety-seven less than a number. Find the number. ______________________________
3. Fifty-seven is 19 more than some number. Find the number. ______________________________
4. A number decreased by 16 is -26. Find the number. ______________________________
5. After buying has 24 more bracelets, Tasha now has 137. How many did Tasha use to have? ______________________________
6. Sarah spent $28.50 of her savings. She now has $42. Previously how much did Sarah have in savings?
______________________________
~~ Unit 2, Page4 ~~
Hint for #14: simplify left side first
HOMEWORK:
Solve each equation:
1. m − 17 = −8 2. k − ½ = 1¼ 3. −44 + n = 36
4. −36 = p − 91 5. m − 21.1 = −36.6 6. 19 = c − (−12)
7. x + 14 = 21 8. 31 = p + 17 9. −19 = k + 9
10. n + 4.7 = −4.7 11. 36 + n = 75 12. −176 = h + (−129)
13. −88 + z = 0 14. −33 + (−7) = 29 + m 15. t + (−2) = −66
Write an equation for each sentence. Solve the equation. 16. The sum of -25 and a number is -73. Find the number.
17. A number increased by 46 is 22. Find the number.
18. Twenty-two less than a number is -85. Find the number
Equation: ___________________
(show work)
Solution: ___________________
Equation: ___________________
(show work)
Solution: ___________________
Equation: ___________________
(show work)
Solution: ___________________
~~ Unit 2, Page5 ~~
SOLVING EQUATIONS BY MULTIPLYING OR DIVIDING
DIVISION PROPERTY OF EQUALITY:
You can divide the same non-zero number from each side of an equation.
Arithmetic Algebra
6 = 3(2) If a = b and c≠0, then
MULTIPLICATION PROPERTY OF EQUALITY:
You can multiply the same number to each side of an equation.
Arithmetic Algebra
12 = 3(4) If a = b, then
12*2 = 3(4)*2 a * c = b * c
EXAMPLES:
a) Dividing to solve an equation: Check your solution.
b) Multiplying to solve an equation: Check your solution.
c) 288 is the product of 12 and a number.
Write an equation. Check your answer.
Solve.
Objectives: I can solve one-step equations using multiplication and division.
~~ Unit 2, Page6 ~~
PRACTICE:
Solve each equation. Check each solution for reasonableness.
a. b. c.
d. e.
f.
g. -30 =
h.
i.
Write an equation, then solve.
j. The product of a number and -4 is 240. What is the number?
k. The quotient of a number and 20 is 40. What is the number?
~~ Unit 2, Page7 ~~
Homework:
Solve each equation. Check each solution for reasonableness.
1. 2. 3.
4. 5.
6.
7.
8.
9.
10. 11. 12.
13. 14.
15.
Write an equation, then solve.
16. Twice a number is . What is the number?
17. The quotient of a number and is . What is the number?
~~ Unit 2, Page8 ~~
FRACTIONAL COEFFICIENTS AND SOLUTIONS
Sometimes you will have a fractional coefficient. (A coefficient is the number multiplied by the
variable.) Sometimes your answers will be a fraction as well. Leave your answers as fractions.
Examples...
1.
2. 5x = 52
x =
Practice.
a. 2p = 5 b. 46 = 4y c.
d. 65 = 20t e.
f.
y = 11
g. 8 + d = 13 h. e 4 = 2 i. 44 =
Objectives: I can solve one step equations
with fractional coefficients and solutions.
Instead of dividing by the
coefficient...multiply by
the reciprocal.
What can you
multiply
by that
will equal 1?
Cross-cancel if possible. 10
1
Divide both sides by 5.
Reduce if possible. Improper fractions do NOT
HAVE to be changed to mixed numbers.
~~ Unit 2, Page9 ~~
Homework:
Solve each equation.
1.
2. 3.
4. 5.
6.
7.
8.
9.
10. 11. 12.
13. 14.
15.
Write an equation, then solve.
16. Two-thirds of a number is -36. What is the number?
17. The product of a number and -3 is -20. What is the number?
~~ Unit 2, Page10 ~~
SOLVING TWO STEP EQUATIONS
Now we'll solve some more complicated equations and inequalities - ones that have two-
step solutions, because they involve two operations. Solving equations is like solving a
puzzle. Just keep working through the steps until you get the variable you're looking for
alone on one side of the equation. This is called isolating the variable.
Here's a two-step equation. Let's start with the variable x, and describe, step by step, what
is being done to x in an equation.
3x - 10 = 14 Equation
3x First, x is ___________ by three.
3x - 10 Next, ten is ___________ from the term ____.
3x - 10 = 14 We get a result of 14.
Start with x --> Multiply by 3 --> Subtract 10 -->
Result is 14.
Solving an equation is like working the equation backwards to discover what number will
work in the equation. Now let's work backwards and use inverse operations to undo all the steps. We can start with the result of 14.
14 Start with result.
14 + 10 Next, working backwards, we can ____ 10, which
is the inverse of _______________ 10.
14 + 10
3
Now we __________ by 3, since that's the
inverse of _______________ by 3.
24 = 8
3 We get an answer of 8.
Start with result of 14 —> Add 10 —> Divide by 3
Answer is 8.
Do you see how it's important when solving an equation to "undo" all the steps in the
correct order? No matter how many steps are in the original equation, you can work backwards and apply the inverse operations, in order, to arrive at the solution!
Solving a two-step equation requires the same procedure(s)
as a one-step equation.
However, the order in which the procedures are done
makes a difference.
Do the inverse operation for
addition or subtraction first.
Do the inverse operation of
multiplication or division last.
Objectives: I can solve two step equations.
~~ Unit 2, Page11 ~~
Step 1: change subtraction to adding the opposite, then add 9 to both sides Step 2: Then divide both sides by 5
5x – 9 = 31 Check your answer.
Perform the inverse operations when solving equations.
When checking your solution, use the correct order of operations. [P E MD AS]
You can use two-step equations to solve many problems. Write an equation for each and solve.
1. Five more than 2 times a number is -7.
Equation: _____________________
Solve Check
2. Two diminished by -7 times a number is 51.
Equation: _____________________
Solve Check
3. Six meters less than twice length x is 18.
Equation: _____________________
Solve Check
~~ Unit 2, Page12 ~~
Homework is continued on the next page
Practice:
Solve and check each solution.
1.
check 2. check
3.
check 4. check
Homework:
Solve and check each solution.
1. check 2. check
3.
check 4.
check
5. check 6.
check
~~ Unit 2, Page13 ~~
7. check 8.
check
9.
check 10.
check
11. Five times a number, decreased by 2 is -37.
Equation: _____________________
Solve Check
12. Six years older than twice age x is 38.
Equation: _____________________
Solve Check
13. Multiple choice: Choose the correct equation for the situation.
50mph slower than twice the speed x is 100mph.
A. 50 + 2x = 100 B. 2x – 50 = 100 C. 50 = 2x + 100 D. 50 – 2x = 100
~~ Unit 2, Page14 ~~
Homework is continued on the next page
Review Solving One and Two Step Equations
Solve and CHECK each problem. SHOW ALL STEPS!!!!!!!
Solve Check
– Solution:
Equation: –
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
– Solution:
Equation: –
Substitution:
Solution:
Equation:
Substitution:
#1
#2
#3
#4
#5
Objectives: I can solve one and two step
equations.
~~ Unit 2, Page15 ~~
Homework is continued on the next page
Solve Check
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
#6
#7
#8
#9
#10
~~ Unit 2, Page16 ~~
Homework is continued on the next page
Solve Check
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
#11
#12
#13
#14
#15
~~ Unit 2, Page17 ~~
Solve Check
- 8 = 4x Solution:
Equation:
Substitution:
x – (-10) = - 3 Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
Solution:
Equation:
Substitution:
#16
#17
#18
#19
#20
~~ Unit 2, Page18 ~~
Assignment is continued on the next 2 pages with an ERQ
SOLVING EQUATIONS
Substitute the given value of x to find the value of y. Find the value of x when given the value of y.
a) Example: b) Example: Solve for y: 3x + 8 = y if x=5 Solve for x: 3x + 8 = y if y=-10 3x + 8 = -10 3(5) + 8 = y - 8 -8 15 + 8 = y 3x = -18
23 = y
=
y = 23 x = - 6
1. a) 2x – 4 = y if x = 14 b) 2x – 4 = y if y=10 2. a) 5m + 12 = n if m = 2 b) 5m + 12= n if n=27 3. a) 3j – 5 = k if j = 6 b) 3j - 5 = k if k=25
4. a) 6u – 3 = t if u=4 b) 6u – 3 = t if t = 27
y is already isolated,
just simplify the
expression
x is not isolated,
you must solve the
equation
Objectives: I can solve one and two step
equations.
~~ Unit 2, Page19 ~~
Read all parts of the extended-response question before you begin. Write your answers to the extended-
response question on the answer page. For each extended-response question, use the grid provided to
create any required charts or graphs. If a question does not require a chart or graph, write your written
response over the grid lines.
The ski club is planning a trip for winter break. They wrote the equation C = 200 + 10n to estimate the cost in dollars of the trip if n students attended. Duncan and Seth both used the equation to estimate the cost for 50 students. Duncan said the cost would be $10,500 and Seth said it would be $700.
a) Determine which estimate is correct. Show the calculations needed to find the estimate. b) What mathematical operations do you need to perform to calculate the cost of the trip? In what order must you perform the operations? c) How do you think Duncan and Seth found such different estimates if they both used the same equation? d) The president of the ski club sent in a check for $950. How many students signed up to go if this is the total cost? Show all calculations.
BE SURE TO LABEL YOUR RESPONSES (a), (b), (c) AND (d).
Rubric Scoring Guide a) ½ point: Determining the correct estimate for 50 students
½ point: Accurately and completely showing the calculations
b) ½ point: Stating the correct operations needed
½ point: Stating the correct order of these operations
c) ½ point: Clearly stating how Duncan and Seth arrived at different estimates
d) ½ point: Determining the correct number of students
1 point: Accurately and completely showing the calculations (½ point for a minor error)
This problem is worth 10 points. Score of 4: You will have a 10/10. Score of 3: You will have a 9/10. Score of 2: You will have a 7.5/10. Score of 1: You will have a 5/10. Score of 0: You will have a 1/10. No attempt: You will have a 0/10
~~ Unit 2, Page20 ~~
Homework is continued on the next page
~~ Unit 2, Page21 ~~
Solve. You must show your work as demonstrated in class!
1. – – 2. – – 3. –
4.
– 5. – 6. –
7.
8. – 9. – –
10. – 11. –
12. – –
13. 14. 15.
16. 17. 5
x + 4 = 19 18.
~~ Unit 2, Page22 ~~
MORE WITH FRACTIONS, FRACTIONAL COEFFICIENTS, AND SOLUTIONS You may have to solve an equation with a big fraction bar that we encountered when we were
simplifying expressions. Remember that when solving equations, you are UNDOING the order of
operations to isolate the variable.
For Example:
Remember, if you have a fractional coefficient, multiply by the reciprocal. You can expect some
problems in the section to have rational (fractional) answers.
Practice:
1)
2)
3)
4)
5)
6) 6p
Objectives: I can solve one and two step equations
with fractions, fractional coefficients and solutions.
First, multiply each side by 5…you can think of the x+4 as being
grouped together in parentheses.
The 5's on the left side cancel out, leaving a one-step equation.
Add (-4) to both side to isolate the variable.
~~ Unit 2, Page23 ~~
HOMEWORK: Solve each equation showing steps as demonstrated in class.
1)
2) 3)
4)
5)
6)
7) 8) 9)
10)
11)
12)
13)
14)
15)
~~ Unit 2, Page24 ~~
SOLVING WORD PROBLEMS USING EQUATIONS When a rate and a starting amount are given in a word problem, a similar equation can usually
be written and solved.
Examples: Write an equation for each situation. Do NOT solve yet!
Sometimes the total is unknown and therefore it will be assigned the variable.
1) A plumber charges $25 for a service call plus $50 per hour of service. Write an equation for
the cost, C, for 2 hours and 30 minutes.
___________________________
2) Nick collected 100 pounds of aluminum cans to recycle. He plans to collect an additional 25
pounds each week for 2 months. (assume four weeks for each month) Write the equation for
the total pounds, P, of aluminum cans.
___________________________
Sometimes there is an amount to be determined and therefore it will be assigned the
variable.
3) For babysitting, Nicole charges a flat fee of $10, plus $5 per hour. Write an equation if Nicole
wants to make a total of $50 after h hours of babysitting.
___________________________
4) Suppose that the water level of a river is 34 feet and that it is receding at a rate of 0.5 foot
per day. Write an equation for the water level after d days to determine how many days will the
water level be 26 feet.
___________________________
Objectives: I can write equations to model real
world problems.
Total = Start Amount + Rate * How Many
A Rate The following are examples of a rate
$3 per day
$2 an hour
60 mph
Per is a key word that is often associated with rate.
Starting Amount A flat fee or starting value.
This value is a constant. It never changes.
~~ Unit 2, Page25 ~~
Homework is continued on the next page
HOMEWORK:
Write an equation for each of the following.
1) A canoe rental service charges a $10 transportation fee and $20 dollars an hour to rent a canoe. Write an equation representing the cost, C, if you are renting the canoe for 6 hours.
___________________________
2) A video rental store charges a $20 membership fee and $2.50 for each video rented. Write an equation to model this situation if a new member paid the store $67.50 in the last 3 months, to find how many videos, v, were rented.
___________________________
3) An attorney charges a fixed fee on $250 for an initial meeting and $150 per hour for all hours worked after that. Write an equation to find the charge, C, for 26 hours of work.
___________________________
4) A water tank already contains 55 gallons of water when Baxter begins to fill it. Water flows into the tank at a rate of 8 gallons per minute. Write an equation to model this situation to find the volume of water in the tank, V, 25 minutes after Baxter begins filling the tank.
___________________________
5) Casey has a small business making dessert baskets. She estimates that her fixed weekly costs for rent and electricity are $200. The ingredients for one dessert basket cost $2.50. Write an equation if her total costs were $562.50 to find how many dessert baskets, d, she made.
___________________________
6) Tim buys a new computer for his office for $1200. For tax purposes, he declares depreciation (loss of value) of $200 per year. Let d be the declared value of the computer after 5 years. Write an equation to
model the value of the computer over time.
___________________________
7) Brenna works as waitress. She earns an hourly wage of $6 plus tips. Today she worked h hours and
was paid a total of $ 108.00. She received $48 in tips. Write an equation to determine how many hours Brenna worked in today's shift.
___________________________
8) The cost of a school banquet is $95 plus $15 for each person attending. Write an equation that
determines the total cost, c, for 77 people.
___________________________
9) A sunflower in Julia's garden was 12 centimeters tall when it was first planted. Since then, it has grown
approximately 0.6 centimeters per day. Write an equation if the sunflower's height is now 102 cm to determine the number of days, d, since it was planted.
___________________________
~~ Unit 2, Page26 ~~
10) Jeanette paid $150 to join a handball club. She pays an additional $15 every time she uses one of the club's handball courts. Write an equation to find the number of times she has played, p, if Jeanette's total
cost for playing handball is $225.
___________________________
REVIEW: Solve each equation showing steps as demonstrated in class.
~~ Unit 2, Page27 ~~
Homework is continued on the next page
SOLVING WORD PROBLEMS USING EQUATIONS Part 2 Now that you have practiced writing equations to model situations, you will also find the solution to the
problems by solving the equations.
Write an equation to model each situation and then solve to find the solution.
Homework
Write an equation to model each situation and then solve to find the solution.
Objectives: I can write and solve equations to model
real world problems.
Equation: ____________________
Work
Solution ____________________
(Include units)
1) Billy plans to paint baskets. The paint
costs $14. The baskets cost $7each.
Write an equation that finds the total cost,
C, if 6 baskets were made. Determine the
cost of six baskets.
2) Felicia paid $125 to join a tennis club. She
pays an additional $5 every time she uses one
of the club's tennis courts. Write an equation
to model this situation to determine n number
of times Felicia played tennis if her total cost
for playing tennis is $300. How many times
did she play?
Equation: ____________________
Work
Solution ____________________
(Include units)
1) A water tank already contains 70 gallons of
water when Ryan begins to fill it. Water flows into
the tank at a rate of 6 gallons per minute. Write an
equation to model this situation to find the volume
of water in the tank, V, 30 minutes after Ryan
begins filling the tank. Determine the final volume.
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
2) Jose works as server in a restaurant. He earns
an hourly wage of $4 plus tips. Today he worked h
hours and was paid a total of $108. He received
$62 in tips. Write an equation to determine how
many hours, h, Jose worked in today's shift.
~~ Unit 2, Page28 ~~
Homework is continued on the next page
4) Rebekah has a small business making dessert
baskets. She estimates that her fixed weekly costs
for rent and electricity are $250. The ingredients
for one dessert basket cost $3.00. Write and solve
an equation if her total costs were $325 to find how
many dessert baskets, d, she made.
3) Shakira buys a new printer for her office for
$800. For tax purposes, she declares depreciation
(loss of value) of $100 per year. Let d be the
declared value of the printer. Write and solve an
equation to model the value of the computer after 5
years.
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
5) A kayak rental service charges a $12
transportation fee and $25 dollars an hour to rent a
canoe. Write and solve an equation representing
the cost, C, if you are renting the kayak for 4 hours.
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
6) A tree in Abram's garden was 22 cm tall when it
was first planted. Since then, it has grown
approximately 7 cm per year. Write an equation and
solve if the tree's height is now 78 cm to determine
the number of years, y, since it was planted.
7) Members of the soccer team are walking to raise
money for a local shelter. Hanna asked her mom
for a $20 donation then each of her friends donated
$2.50 each. Write and solve an equation to
determine how many of her friends, f, donated if
she raised a total of $40.
8) Beaumont Bowling Lanes charges a flat fee of
$15 plus $2 per game. Write and solve an equation
to calculate the total bill, T, if Jonea and Casey
played 4 games.
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
~~ Unit 2, Page29 ~~
9) A day camp charges a $35 fee for activities plus
$150 per week. Write and solve an equation to
model this situation to find the number of weeks
Lane attended camp if the total owed is $935.
10) The roller rink charges an admittance fee of $6
plus $3 per hour. Write and solve an equation to
calculate how many hours, h, Shareef skated if he
paid a total of $20.
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
11) A car rental charges a flat fee of $40 plus $12
per day. What is your total charge, C, if you
needed the car for five days? Write an equation
and solve.
Equation: ____________________
Work
Solution ____________________
(Include units)
12) Pi Pizza charges a $5 delivery charge and $8
per large pizza. If the total cost is $37, write and
solve an equation to determine how many pizzas
were ordered, p.
13) Tristan is pouring sand into a cylinder at a rate
of 1.5 pounds/minute. The cylinder started with 8
Lbs of sand before he started. How much sand did
the cylinder hold if it took Tristan 10 minutes to fill
it? Write and solve an equation using s for the
volume of the cylinder.
14) A new candle is 15 inches tall. If it burns at a
rate of 1/2 inch per hour, how long will it take for
the candle to burn out? (Height would be zero.)
Write and solve an equation using t for the number
of hours.
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
Equation: ____________________
Work
Solution ____________________
(Include units)
~~ Unit 2, Page30 ~~
Solving Multi-Step Equations
Sometimes one side of an equation will need to be simplified before you can solve. You may
have to combine like terms or use the distributive property.
Practice using the distributive property and combining like terms with these expressions:
Now we'll simplify one side of the equation before we solve.
8. 9. 10.
11. 12.
Objective: I can solve multi- step equations with the
distributive property and combining like terms.
First...Distribute if possible
Second...Combine like terms if
possible
Third...isolate the variable by
UNDOING the order of operations.
~~ Unit 2, Page31 ~~
Practice
13. 14.
Homework
1. 2. 3.
4. 5. 6.
7. 8.
9. 10.
~~ Unit 2, Page32 ~~
VARIABLE TERMS ON BOTH SIDES
To solve when you have variables on both sides, eliminate one of these terms by adding the
opposite of it to both sides.
Examples
1. 5x + 6 = 2x +15 2. 3.
Practice.
1. 2. 3.
4. 5. 6.
Write an equation and solve. Let "x" = the number.
7. Twenty decreased by 2 times a number is the same as 10 less than 3 times the number. Find
the number.
Equation: _____________________
Solve
Objective:I can solve multi- step equations
with variables on both sides.
Add -2x to
both sides first
Make sure that you only combine LIKE
terms. Line them up accordingly.
~~ Unit 2, Page33 ~~
Homework is continued on the next page
Homework.
1. 2. 3.
4. 5. 6.
Review.
7. 8. 9.
10.
11.
12. -8k +12 = -32
~~ Unit 2, Page34 ~~
Write an equation and solve. Let "x" = the number.
13. Eight less than 7 times a number is the same as 4 more than 3 times the number. Find the
number.
Equation: _____________________
Solve
14. Four more than 6 times a number is the same as 9 times the number increased by 10. Find
the number.
Equation: _____________________
Solve
15. A number plus 5 more than 3 times the number is 37. Find the number.
Equation: _____________________
Solve
~~ Unit 2, Page35 ~~
NO SOLUTION AND ALL REAL NUMBERS
Examples:
1.
2. 3.
Practice:
1. 2. 3.
4. 5. 6.
Objectives: I can solve multi- step equations with
variables on both sides with the distributive property
and Identify equations with no solution or all real
numbers.
If you get a TRUE statement...like 5=5,
then all real numbers would satisfy
the equation...write ALL REAL #'s
If you get a FALSE statement...like
5=2, then no solution would satisfy
the equation...write NO SOLUTION
~~ Unit 2, Page36 ~~
Homework.
1. 2.
3. 4.
5. 6.
Write an equation and solve. Let "x" = the number.
7. Four times a number is the same as 14 less than twice the number. Find the number
Equation: _____________________
Solve
8. One more than 8 times a number is the same as 12 times the number decreased by 3. Find
the number.
Equation: _____________________
Solve
~~ Unit 2, Page37 ~~
INEQUALITIES AND THEIR GRAPHS Equations
We have been studying equations and how to solve them. Let’s review. What is an equation? An equation is: ____________________________________________________________________ ________________________________________________________________________________ To find a value for the variable that makes the equations true, we SOLVE the equation. Any value for the
variable that makes an equation true is called the of the equation.
Inequalities An inequality is a mathematical sentence that shows the relationship between quantities that are not
equal, using inequality symbols.
Some students have trouble remembering which symbol is greater than and which one is less than. Do
you have a trick to help them remember?
Examples of Inequalities
To find a value for the variable that makes the inequalities true, we SOLVE the inequality. Any value for
the variable that makes an inequality true is called the of the inequality.
Graphing Solutions of Inequalities
Example #1
What values for y would make this inequality true? ___________________________________________
Is 3 a solution for this inequality? Explain. _________________________________________________
An _____________dot shows that 3 is NOT a solution. Shade all the points to the left of 3 since y is less
than 3.
Objectives: I can graph solutions of
inequalities and write inequalities to
describe real-world situations.
~~ Unit 2, Page38 ~~
Example #2
How can we rewrite this inequality so that it is still true but the variable is first? _____________________
What values for n would make this inequality true? ___________________________________________
Is –2 a solution for this inequality? Explain. _______________________________________________
An _______________dot shows that –2 is NOT a solution. Shade all the points to the right of –2 since n
is greater than –2.
Example #3
What values for w would make this inequality true? ___________________________________________
Is –3 a solution for this inequality? Explain. ________________________________________________
A __________________dot shows that –3 is a solution. Shade all the points to the right of –3 since w is
greater than –3.
Example #4
How can we rewrite this inequality so that it is still true but the variable is first? _____________________
What values for k would make this inequality true? ___________________________________________
Is 9 a solution for this inequality? Explain. _________________________________________________
A ___________________ dot shows that 9 is a solution. Shade all the points to the rightt of 9 since k is
greater than 9.
~~ Unit 2, Page39 ~~
Writing Inequalities to Describe Graphs
_______________ _______________
Writing Inequalities to Describe Real-World Situations
You can write inequalities to describe real-world situations. What inequality symbols do you think you will
use when you see the following phrases:
Write inequalities to represent these real-life situations.
1) Amusement Parks make rules for safety. The sign the
amusement park (in the picture at the right) tells you how tall
you have to be to ride. Write an inequality for this situation.
Let h be height in feet.
Words
Inequality
2) You must be at least 17 years old to get into an R-rated movie. _______________________ 3) The room can hold at most 210 people. ___________________________ 4) No more than 7 people can fit into the SUV. _______________________ 5) You must be at least 16 years old to get your driver’s license. ________________________ 6) There can be no less than 30 chairs in the classroom. ________________________ 7) We need no less than 550 hot dogs for the food competition. _______________________
at most
no more than
at least
no less than
~~ Unit 2, Page40 ~~
Homework is continued on the next page
Homework
Write an inequality for each graph. Name your own variable.
~~ Unit 2, Page41 ~~
Review
Simplify.
21. – ____________ 22. ____________
23. ____________ 24. ____________
Solve.
25.
26. 9d + 7 = 19 27.
28. 29. 30.
~~ Unit 2, Page42 ~~
SOLVING ONE-STEP INQUALITIES
>
Solving linear inequalities is pretty much the same as solving equations.
Don’t panic!
<
Note: Inequalities have more than one solution!
Example #1 --
Solve and graph the inequality.
–
Example #2 --
Solve and graph the inequality.
Example #3 --
Solve and graph the inequality.
Example #4 --
Solve and graph the inequality.
Objectives: I can solve and graph one-
step linear inequalities including real-
world situations.
Remember we said, solving linear inequalities is pretty much the same
as solving equations? There is a VERY IMPORTANT exception!
~~ Unit 2, Page43 ~~
Addition
Look at this true statement:
Suppose we add 3 to each side.
What is the relationship between these two numbers?
Is the relationship the same as before you added 3 to each side? ____________
Subtraction
Look at this true statement:
Suppose we subtract 7 from each side. – –
What is the relationship between these two numbers?
Is the relationship the same as before you subtracted 7 from each side? ____________
Multiplication
Look at this true statement:
Suppose we multiply both sides by .
What is the relationship between these two numbers?
Is the relationship the same as before you multiplied by ? ____________ If not, how do we “fix” the
When you multiply or divide an inequality by a negative
number, it changes the direction of the inequality symbol!
Let’s try these together…
1) 2)
3)
4)
5) Write an inequality for the situation. Then solve and graph the inequality.
You must be at least 48 inches tall to ride an amusement park ride, and your little
sister is 39 inches tall. How many inches (i) must she grow before she can ride the
ride?
~~ Unit 2, Page45 ~~
Homework is continued on the next page
Homework
Solve and graph.
1) 2)
3) 4)
5)
6)
7)
8)
9) – 10)
~~ Unit 2, Page46 ~~
11) 12)
Write an inequality for the situation. Then solve the inequality.
13) You need no more than 3000 calories in a day. You consumed 840 calories at breakfast and
1150 calories at lunch. How many calories (c) can you eat for dinner?
14) I can spend $65 at the most at the mall today. I spent $23 at The Gap and $30 at Rue 21.
How much more (d) can I spend today?
Review: Solve each equation.
15)
16)
17)
18)
19) 20)
19) 20)
~~ Unit 2, Page47 ~~
SOLVING TWO STEP INEQUALITIES
Solving two-step inequalities is pretty much the same as solving two-step equations with a very important exception---when you multiply or divide an inequality by a negative number, it changes the
direction of the inequality symbol!
Examples:
Solve and graph:
#1 #2
+ 3 > 11
#3 Write an inequality and solve.
You divide a number x by -3. Then you subtract 1 from the quotient. The result is at most 5. Write and
solve an inequality to find all possible solutions.
Inequality: _____________________________
#1
Objectives:I can solve and graph two-step linear
inequalities including writing and solve
inequalities for real-world situations.
c.
~~ Unit 2, Page48 ~~
#2) Write and inequality for the situation. Then solve the inequality.
You multiply a number, , by . Then you add 16 to the product. The result is at most .
Inequality: ____________________________________
Solve
& graph.
Homework:
Solve and graph.
1) 2)
3)
4)
5) 6)
~~ Unit 2, Page49 ~~
7)
8)
9) 10)
Write an inequality for the following two situations. Then solve.
11) On a trip from Kentucky to Florida, your family wants to travel at least 420 miles in 6 hours of
driving. Write and solve an equation as to what your average speed must be.
12) You want to spend at most $20 for a taxi ride. Before you go anywhere, the driver sets the meter at
the initial charge of $5. The meter then adds $2 for every mile you drive. What is the farthest you can
go?
13) Error Analysis A student solved
and graphed the inequality
What error did the student
make? (Justify your answer.)
~~ Unit 2, Page50 ~~
Homework is continued on the next page
Review: Solving Equations and Inequalities
Solve each of the following. Show all steps. Leave all answers as integers or fractions
in simplest form. Do not round. Put a rectangle around your answers.
1) 2)
3) 4) –
5) 6)
7) 8)
9) 10)
11) –
12)
13)
Simplify by combining like terms.
14) – 15) – 16) –
Use the distributive property to multiply. Simplify if possible.
17) 18) 19) – 20)
~~ Unit 2, Page51 ~~
Homework is continued on the next page
Solve each of the following. Show all steps. Leave all answers as integers or fractions
in simplest form. Do not round. If your solution is "all real numbers" or "no solution,"
state that. Put a rectangle around your answers.
21) 22) 23) –
24) – – 25) –
26)
27)
28)
~~ Unit 2, Page52 ~~
Solve and graph. Work must be shown.
29) – 30) 31)
32) – – 33) – – – 34)
– 5
35) Choose the best inequality to describe the situation. You must be at least 18 years old
to vote.
A. v 18 B. v < 18 C. v > 18 D. v 18 36) Choose the correct inequality to describe the situation. The concert tickets are less than $50.00 each.
A. c $50.00 B. c <$50.00 C. c > $50.00 D. c $50.00